since they are particularly important in applications
[1]}. Moreover, in some important applications, the leading coefficient \(P_2\)  of \(P(\lambda )\)  has low rank [2]}, a property related in the regular case to the presence of an infinite eigenvalue with high multiplicity. This presence is a challenge for the best available algorithms that compute all the eigenvalues of a regular dense quadratic polynomial matrix [3]}, [4]}, since infinite eigenvalues have to be deflated. The algorithms in [3]}, [4]} are based on the GLR-strong linearization known as the second Frobenius companion form, but the strongly minimal linearization constructed in Theorem  might be a competivite option since, as we show in this section, it is very simple and much smaller than the Frobenius companion form for quadratic polynomial matrices with \(P_2\)  having low rank.
